There are a lot of people that have cultural/religion roots that mandates them to be very fervorous in praising God the most that they can. As other answers already stated, this is common in Middle East, but I already saw some Christian people in the West with the same trait (in my own family to be sincere).
For those people, not praising God wherever they can is bad, ranging from uncomfortable, to simply wrong, to unethical or to sinful, regardless of what the receiver of the message feels about that. Their religious belief mandates that they should praise God wherever possible.
Since you should value inclusiveness, you should not judge people that have such religious views either positively nor negatively. You might personally consider this as fanatism, lunacy, insanity or whatever, but you shouldn't allow your opinion about the candidate's religious belief contaminate your judgement.
To be frank, I also consider that those people are fanatics, and I have some of them in my own family. Also, most of them even are very aware that everyone else considers them as fanatics and they're very proud of this because this makes them believe that they are being successfully in pleasing God if everyone else is seeing them this way. They surely can be very annoying and boring with their religious views and behaviors and they are perfectly aware of that, but they strongly believe that it is an important mandatory part of the mission of their lives acting in such way. So, they are simply behaving in the way that they believe that they should behave regardless of what other people thinks about that.
As any religious belief, their way of life should be respected and tolerated, not suppressed or discriminated. Even if you find that they can behave in a somewhat annoying way to you and to everyone else.
So I made it “set in stone” that politics and religion do not belong in the lab.
Most people would be ok with that. But for those “overly-fervorous”, this sound as religious persecution, suppression or censorship precisely because their religious belief says that they should praise God and talk about their religion as much as possible to whoever possible wherever possible even if the audience does not wants to hear them. In this case, as long as they tolerate other people and doesn't creates serious troubles about religious issues with others, everyone for the sake of inclusiveness should allow them be what they are - fervorous religious.
Inclusivity and respect is not achieved by silencing everyone, even if it is about an issue that have nothing to do with the work or the environment that you're working on. It is achieved by having everyone be ok with everyone else speaking about things that no one else believes (as long as it is not dishonest, unethical, criminal, illegal or something like that).
So, change the rule to the similar, but quite different and somewhat weaker as:
“We prefer to not talk in the lab about things that does not belong to the lab as long as possible, like, for example, sports, politics, sexuality and religion.”
You should just ensure that no one creates trouble about that, that everyone knows the rules and that everybody respects everyone else. Just be sure that the rules don't have the unintended effect to lead to some form of censorship, suppression of free speech or something like that. This is something that have a very tenuous line and a lot of shady areas, but as long as everyone gets and understand the rules, it shouldn't create serious issues having someone saying the word "God" too much because they believe that its their obligation to do so.
To directly answer your questions:
Is it acceptable to disclose and declare vividly religious preferences in academic applications?
Yes. This is not the ideal, surely, but should not be disallowed. Disallowing that is a form religious persecution and uninclusiveness.
Is this a sign that this person will also be as vocal in the group, thus creating issues?
Probably. But you can't act against someone due to some issue that they did not created yet even if you are pretty sure that they eventually will.
Should I simply ignore this information, even if the candidate decided for some reason to vocally disclose it? What should I do with it?
Ignore it and evaluate their C.V. as if that information was not there and then evaluate/score it as you would do with any other C.V.